The Trigonometric Ratios
If \(\theta\) is an angle in standard position, and \((x,y)\) is a point on its terminal side, with \(r = \sqrt{x^2 + y^2}\text{,}\) then
If \(\theta\) is an angle in standard position, and \((x,y)\) is a point on its terminal side, with \(r = \sqrt{x^2 + y^2}\text{,}\) then
To construct a reference triangle for an angle :
Quadrant I: \(~~~~~~\theta = \widetilde{\theta}\) | \(\hphantom{0000}\) | Quadrant II: \(~~~~\theta = 180\degree - \widetilde{\theta}\) |
Quadrant III: \(~~~~~\theta = 180\degree + \widetilde{\theta}\) | \(\hphantom{0000}\) | Quadrant IV: \(~~~~\theta = 360\degree - \widetilde{\theta}\) |
Let \(P\) be a point on a unit circle determined by the terminal side of an angle \(\theta\) in standard position. Then the coordinates \((x,y)\) of \(P\) are given by
If point \(P\) is located at a distance \(r\) from the origin in the direction specified by angle \(\theta\) in standard position, then the coordinates of \(P\) are
The angle of inclination of a line is the angle \(\alpha\) measured in the positive direction from the positive \(x\)-axis to the line. If the slope of the line is \(m\text{,}\) then
where \(0\degree \le \alpha \le 180\degree\text{.}\)
The function \(y = f(x)\) is periodic if there is a smallest value of \(p\) such that
for all \(x\text{.}\) The constant \(p\) is called the period of the function.
The London Eye, the world's largest Ferris wheel, completes one revolution every 30 minutes. By how many degrees will it rotate in 1 minute?
The London Eye in Problem 1 has 32 cabins evenly spaced along the wheel. If the cabins are numbered consecutively from 1 to 32, what is the angular separation between cabins number 1 and number 15?
For Problems 3–4,find two angles, one positive and one negative, that are coterminal with the given angle.
For the angles in Problems 5–6, state the corresponding quadrant and reference angle. Give three other angles with the same reference angle, one for each of the other three quadrants. Sketch all four angles.
Let \(\widetilde{\theta} = f(\theta)\) be the function that gives the reference angle of \(\theta\text{.}\) For example, \(f(110\degree) = 70\degree\) because the reference angle for \(110\degree\) is \(70\degree\text{.}\)
Fill in the table of values.
\(\theta\) | \(0\degree\) | \(30\degree\) | \(60\degree\) | \(90\degree\) | \(120\degree\) | \(150\degree\) | \(180\degree\) | \(210\degree\) | \(240\degree\) | \(270\degree\) | \(300\degree\) | \(330\degree\) | \(360\degree\) |
\(f(\theta)\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) |
Choose appropriate scales for the axes and graph the function for \(-360\degree \le \theta \le 360\degree\text{.}\)
Let \(\widetilde{\theta} = f(\theta)\) be the function that gives the reference angle of \(\theta\text{.}\) (See Problem 7.) Is \(f\) a periodic function? If so, give its period, midline, and amplitude. If not, explain why not.
For Problems 9–20, solve the equation exactly for \(0\degree \le \theta \le 360\degree\text{.}\)
\(\sin \theta = \dfrac{-1}{2}\)
\(\cos \theta = \dfrac{-1}{\sqrt{2}}\)
\(2\cos \theta + 1 = 0\)
\(5\sin \theta + 5 = 0\)
\(\tan \theta - 1 = 0\)
\(\sqrt{3} + 3\tan \theta = 0\)
\(\cos \theta = \cos(-23\degree)\)
\(\sin \theta = \sin(370\degree)\)
\(\tan \theta = \tan 432\degree\)
\(\tan \theta = \tan (-6\degree)\)
\(\sin \theta + \sin 83\degree = 0\)
\(\cos \theta + \cos 429\degree = 0\)
For Problems 21–26, solve the equation for \(0\degree \le \theta \le 360\degree\text{.}\) Round your answers to two decimal places.
\(3\sin \theta + 2 = 0\)
\(5\cos \theta + 4 = 0\)
\(\dfrac{2}{3}\tan \theta + 1 = 0\)
\(-4\tan \theta + 12 = 0\)
\(4 = 8\cos \theta + 9\)
\(-3 = 6\sin \theta -5\)
For Problems 27–32, find the coordinates of the point where the terminal side of angle \(\theta\) in standard position intersects the circle of radius \(r\) centered at the origin. Round your answers to two decimal places.
\(\theta = 193\degree,~ r = 10\)
\(\theta = -12\degree,~ r = 20\)
\(\theta = 92\degree,~ r = 8\)
\(\theta = 403\degree,~ r = 6\)
\(\theta = -341\degree,~ r = 3\)
\(\theta = -107\degree,~ r = 20\)
If you follow a bearing of \(190\degree\) for 10 miles, how far south and how far west are you from your starting point? Round to two decimal places.
A child releases a balloon, which then follows a bearing of \(84\degree\) for 150 meters. How far east and how far north is the balloon from where it was released? Round to the nearest meter.
For Problems 35–38, write the equation of a sine or cosine function with the given properties and sketch the graph, including at least one period.
Amplitude 7, midline \(y = 4\text{,}\) period 2, \(y\)-intercept 4
Amplitude 100, midline \(y = 50\text{,}\) period 12, \(y\)-intercept 100
Maximum point at \((90\degree, 24)\) and minimum point at \((270\degree, 10)\)
Horizontal intercept at \(180\degree\text{,}\) maximum points at \((0\degree, 5)\) and \((360\degree, 5)\)
For Problems 39–42, evaluate the expression for \(f(\theta) = \sin \theta\) and \(g(\theta) = \cos \theta\text{.}\)
\(2f(\theta)g(\theta)\) for \(\theta = 30\degree\)
\(f(\dfrac{\theta}{2})\) for \(\theta = 120\degree\)
\(g(4\theta) - f(2\theta)\) for \(\theta = 15\degree\)
\(\dfrac{1 - g(2\theta)}{2}\) for \(\theta = 45\degree\)
For Problems 43–46, write an equation for the given graph, and give the exact coordinates of the labeled points.
Every 24 hours Delbert takes 50 mg of a therapeutic drug. The level of that drug in Delbert's bloodstream immediately jumps to its peak level of 60 mg, but the level diminishes to its lowest level of 10 mg just before the next dose.
A water fountain has water trickling into a container, but once the container is full, it tilts and the water pours quickly out. Then the container tilts back and starts to fill again. The container is filled 5 times every minute.
Henry is watching Billie ride a carousel. He stands 2 meters from the carousel, which has a diameter of 10 meters. He notices that she passes by him three times each minute.
An ant walks at constant unit speed (1 unit of distance per second) along the triangle with vertices at \((0,0),~(1,1),\) and \((0,1)\text{.}\)
For Problems 51–54,
\(y = 4 + 2 \cos \theta\)
\(y = -1 + 3 \cos \theta\)
\(y = 1.5 + 3.5 \sin 2\theta\)
\(y = 1.6 + 1.4 \sin (0.5\theta)\)
For Problems 55–58, find the angle of inclination of the line.
\(y = \dfrac{\sqrt{3}}{3} x + 1\)
\(y = - x - 11\)
\(y = 100 - 28 x\)
\(y = - 3.7 + 1.4x\)
For Problems 59–62, find an equation for the line passing through the given point with angle of inclination \(\alpha\text{.}\)
\((0,2)\text{,}\) \(~ \alpha = 45\degree\)
\((4,0)\text{,}\) \(~ \alpha = 135\degree\)
\((3,-4)\text{,}\) \(~ \alpha = 120\degree\)
\((-7,2)\text{,}\) \(~ \alpha = 60\degree\)
Sketch the graphs of \(y = \tan \theta\) and \(y = \cos \theta\) on the same grid for \(-180\degree \lt \theta \lt 180\degree\text{.}\) How are the \(\theta\)-intercepts of the graph of \(y = \cos \theta\) related to the graph of \(y = \tan \theta\) ?
Sketch the graphs of \(y = \tan \theta\) and \(y = \sin \theta\) on the same grid for \(-180\degree \lt \theta \lt 180\degree\text{.}\) How are the \(\theta\)-intercepts of the graph of \(y = \sin \theta\) related to the graph of \(y = \tan \theta\) ?